We also mention that regarding the existence of random attractors in the initial space \(X\)  for the RDS over one parametric space, the good references
are [1]}, [2]}, [3]}, [4]}, [5]}, [6]}. However, here we recall from [7]} some basic notions regarding RDS over two parametric spaces.
Let \(\Omega _1\)  be a nonempty set on which there is a mapping \(\vartheta _{1,t}: \Omega _1\rightarrow \Omega _1\)  such that \(\vartheta _{1,0}\)  is the identity on \(\Omega _1\) 
and \(\vartheta _{1,s+t}=\vartheta _{1,t}\circ \vartheta _{1,s}\)  for all \(s,t\in \mathbb {R}\) . Let \((\Omega _2,\mathcal {F}_2,{P})\)  be a probability space on which there is a measure preserving transformation \(\vartheta _{2,t}\)  such that \(\vartheta _{2,0}\)  is the identity on \(\Omega _2\) 
and \(\vartheta _{2,s+t}=\vartheta _{2,t}\circ \vartheta _{2,s}\)  for all \(s,t\in \mathbb {R}\) . We often use \((\Omega _1,\lbrace \vartheta _{1,t}\rbrace _{t\in \mathbb {R}})\)  and \((\Omega _2,\mathcal {F}_2,{P},\lbrace \vartheta _{2,t}\rbrace _{t\in \mathbb {R}})\)  to denote the two parametric spaces respectively.
